含應(yīng)變梯度效應(yīng)的彈性理論及其應(yīng)用研究
發(fā)布時間：2018-08-09 15:36
【摘要】：微構(gòu)件是微機電系統(tǒng)中的基本構(gòu)件,準確掌握微構(gòu)件的力學行為是實現(xiàn)微機電系統(tǒng)精確控制的基礎(chǔ)。對特征尺寸在微米及亞微米量級的構(gòu)件而言,不論其力學性能還是多場耦合性能均表現(xiàn)出明顯尺寸效應(yīng)。傳統(tǒng)理論不能描述微構(gòu)件力學行為的尺寸效應(yīng),已有研究初步證明考慮高階變形量影響的應(yīng)變梯度理論可以解釋這種尺寸效應(yīng)。然而,現(xiàn)有應(yīng)變梯度理論中,有的尺度參量不獨立,有的引入了不恰當?shù)钠胶鈼l件,有的是近似理論。這些理論描述的微構(gòu)件力學行為的尺寸效應(yīng)均不能真實、合理地反映實際情況。因此,無論是從應(yīng)變梯度理論的發(fā)展來說,還是準確描述微構(gòu)件的力學行為,建立正確合理的有效理論,對微構(gòu)件的力學性能包括其力電耦合性能進行深入研究是當前的重要研究內(nèi)容。本文的研究內(nèi)容主要包括：發(fā)展了應(yīng)變梯度張量的對稱／反對稱和靜水／偏量分解方法,將應(yīng)變梯度張量分解為兩種形式的獨立分量。用應(yīng)變梯度獨立分量重構(gòu)了應(yīng)變梯度彈性理論的本構(gòu)方程,構(gòu)造了各向同性彈性尺度張量。根據(jù)高階張量理論及彈性尺度張量間的約束關(guān)系,從理論上證明了獨立的尺度參量只有3個,解決了應(yīng)變梯度彈性理論本構(gòu)關(guān)系的基本理論問題。進而,提出了尺度參量獨立的應(yīng)變梯度彈性理論,發(fā)展了獨立分量形式的應(yīng)變梯度理論變分原理,推導了控制方程和邊界條件,并給出了正交曲線坐標系下應(yīng)變梯度理論變形量的可用公式。應(yīng)用新發(fā)展的應(yīng)變梯度彈性理論,采用平面應(yīng)變假設(shè),發(fā)展了梁彎曲理論。定義了梁彎曲中的軸力、剪力、彎矩、高階軸力和高階彎矩,并給出了用這些應(yīng)力合力量表達的平衡方程和邊界條件。具體求解了兩種情況下的懸臂梁彎曲問題：一種考慮為平面應(yīng)變問題,另一種考慮為伯努利-歐拉梁問題。兩種情況下的理論分析結(jié)果分別與環(huán)氧樹脂和硅懸臂梁的彎曲實驗結(jié)果相吻合,驗證了新理論的有效性。應(yīng)用新理論與Aifantis單參數(shù)理論分別分析了桿扭轉(zhuǎn)、固定層剪切、薄梁純彎曲和球體膨脹四個典型問題中的尺寸效應(yīng)。通過對比兩種理論下的分析結(jié)果,說明了新理論可統(tǒng)一有效的描述各種變形問題中的尺寸效應(yīng),而單參數(shù)理論具有局限性,揭示了應(yīng)變梯度彈性理論包含多尺度參數(shù)的必要性。針對微機電系統(tǒng)廣泛采用的層復合結(jié)構(gòu),分別提出了適用于層合微梁和微板的位移模式。應(yīng)用新發(fā)展的位移模式和最小勢能原理,建立了雙層微梁和微板的尺寸效應(yīng)模型,推導了相應(yīng)的平衡方程和邊界條件。具體分析了受均布載荷作用的簡支雙層微梁和四邊簡支雙層微板的彎曲問題。分析結(jié)果發(fā)現(xiàn),雙層微梁和微板的彎曲撓度及軸向應(yīng)力均表現(xiàn)出明顯的尺寸效應(yīng)。且當一層微梁或微板厚度遠遠大于另一層微梁或微板厚度時,雙層微梁或微板的撓度變形接近于單層微梁或微板的撓度變形。將尺度參量獨立的應(yīng)變梯度彈性理論拓展到中心對稱介電材料,發(fā)展了只包含3個獨立尺度參量和2個撓曲電耦合參量的撓曲電理論。給出了內(nèi)能密度函數(shù)的具體表達形式以及獨立分量形式的本構(gòu)關(guān)系。發(fā)展了獨立分量形式的撓曲電理論變分原理,推導了控制方程和邊界條件,給出了廣義靜電力的表達形式。新發(fā)展的撓曲電理論同時考慮了尺寸效應(yīng)、極化梯度效應(yīng)和正逆撓曲電效應(yīng)的影響。通過理論推證發(fā)現(xiàn),并非所有的應(yīng)變梯度量都能誘導極化,而是只有膨脹梯度和偏轉(zhuǎn)動梯度的反對稱部分能夠誘導極化,拉伸梯度和偏轉(zhuǎn)動梯度的對稱部分不能誘導極化。應(yīng)用新發(fā)展的撓曲電理論,建立了伯努利-歐拉梁和基爾霍夫圓板的撓曲電效應(yīng)模型,給出了平衡方程和邊界條件。分別針對自由端受集中力和上下表面間受電壓作用的懸臂梁以及受均布載荷和上下表面間受電壓作用的簡支軸對稱圓板,研究了懸臂梁和簡支軸對稱圓板的正逆撓曲電效應(yīng)特性。研究發(fā)現(xiàn),懸臂梁和簡支軸對稱圓板的正逆撓曲電效應(yīng)均表現(xiàn)出明顯尺寸依賴性,且隨著撓曲電耦合系數(shù)的減小而減弱,特別的當撓曲電耦合系數(shù)為零時,正逆撓曲電效應(yīng)消失。本文提出的應(yīng)變梯度彈性理論及撓曲電理論能夠有效預測微構(gòu)件力學性能的尺寸效應(yīng)及力電耦合性能的撓曲電效應(yīng)。研究成果對MEMS產(chǎn)品的設(shè)計分析、性能預測和實驗研究都具有重要的指導意義。
[Abstract]:Micro components are the basic components of microelectromechanical systems. The accurate grasp of the mechanical behavior of the microstructures is the basis for precise control of the microelectromechanical systems. For the components of the size of micrometers and submicrons, the mechanical properties and the multi field coupling performance show obvious size effects. The traditional theory can not describe the force of the micro component. The size effect of learning behavior has been studied. It has been proved that the strain gradient theory considering the influence of high order deformation can explain this size effect. However, in the existing strain gradient theory, some scale parameters are not independent, some have introduced improper equilibrium conditions and some are similar to theory. The size effect can not be true and reflects the actual situation reasonably. Therefore, it is an important research content to study the mechanical behavior of the micro component, whether it is from the development of the strain gradient theory, or to accurately describe the mechanical behavior of the micro component, and to study the mechanical properties of the microstructures, including the mechanical and electrical coupling properties. The main contents include: developing the symmetric / antisymmetric and hydrostatic / partial decomposition method of strain gradient tensor, decomposing the strain gradient tensor into two forms of independent component. The constitutive equation of strain gradient elastic theory is reconstructed with the strain gradient independent component, and the isotropic elastic tensor is constructed. The constraint relation between the theory and the elastic scale tensor proves that only 3 independent scale parameters have been theoretically proved, and the basic theoretical problem of the constitutive relation of strain gradient elasticity is solved. Then, the strain gradient elastic theory with independent scale parameter is proposed, and the variational principle of strain gradient theory is developed, and the theory of strain gradient theory is developed. The equation and boundary condition are controlled and the formula of strain gradient deformation in the orthogonal curvilinear coordinate system is given. Using the new developed strain gradient elasticity theory and the assumption of plane strain, the beam bending theory is developed. The axial force, shear, bending moment, high order axial force and high order bending moment are defined in the bending of the beam, and the use of these stresses is given. The equilibrium equations and boundary conditions expressed by force force are solved in two cases: one is the problem of plane strain and the other is the Bernoulli Euler beam problem. The theoretical analysis results in two cases are in agreement with the experimental results of the bending of epoxy resin and silicon cantilever beam respectively. The new theory and Aifantis single parameter theory are applied to analyze the size effects of four typical problems: rod torsion, fixed layer shear, pure bending of thin beams and spheroid expansion. By comparing the analysis results under the two theories, it shows that the new theory can describe the size effect of various deformation problems in a unified and effective way, and the single parameter is single parameter. The theory of number is limited, which reveals the necessity of multi scale parameters in the strain gradient elasticity theory. According to the laminated structure widely used in microelectromechanical systems, the displacement modes suitable for laminated microbeams and microplates are put forward respectively. The size effect of double microbeams and microplates is established by applying the newly developed displacement mode and the principle of minimum potential energy. The corresponding equilibrium equations and boundary conditions are derived, and the bending problems of simple supported double layer microbeams and four simply supported double-layer microplates are analyzed. The results show that the bending deflection and axial stress of the double-layer microbeam and the micro plate show a clear scale effect, and the thickness of a layer of micro beam or the microplate is far from the same. The deflection deformation of the double layer microbeam or microplate is close to the deflection and deformation of the single layer microbeam or the microplate when the thickness of the micro beam or microplate is larger than that of the other layer. The strain gradient elastic theory which is independent of the scale parameter is extended to the central symmetric dielectric material, and the flexure theory, which contains only 3 independent scale parameters and 2 flexural coupling parameters, is developed. The specific expression of the internal energy density function and the constitutive relation of the independent component form. The variational principle of the flexural theory of the independent component is developed, the control equation and the boundary condition are derived, and the expression of the generalized static electricity is given. The new developed flexure theory takes into account the size effect, the polarization gradient effect and the positive inverse. It is found by theoretical evidence that not all strain gradient quantities can induce polarization, but only the anti symmetric part of the expansion gradient and partial rotational gradient can induce polarization, and the symmetric part of the tensile gradient and partial rotational gradient can not induce polarization. The new developed flexo theory has been used to establish Bernoulli Europe. The balance equation and boundary condition are given for the flexural effect model of the tension beam and the Kirchhoff circular plate. The positive inverse of the cantilever beam and the simply supported axisymmetric circular plate are studied for the cantilever beam of the free end subjected to the concentrated force and the voltage acting on the upper and lower surfaces, as well as the simple supported axisymmetric circular plates subjected to the voltage action between the uniform load and the upper and lower surfaces. It is found that the positive and inverse flexure effect of the cantilever beam and the simply supported axisymmetric circular plate shows a significant dimension dependence, and decreases with the decrease of the flexure coupling coefficient, especially when the flexure coupling coefficient is zero. The strain gradient elasticity theory and the flexure electricity proposed in this paper are proposed. The theory can effectively predict the size effect of the mechanical properties of microstructures and the flexural effect of the mechanical and electrical coupling properties. The research results have important guiding significance for the design analysis, performance prediction and experimental research of MEMS products.
【學位授予單位】：山東大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：TH-39
本文編號：2174545
[Abstract]:Micro components are the basic components of microelectromechanical systems. The accurate grasp of the mechanical behavior of the microstructures is the basis for precise control of the microelectromechanical systems. For the components of the size of micrometers and submicrons, the mechanical properties and the multi field coupling performance show obvious size effects. The traditional theory can not describe the force of the micro component. The size effect of learning behavior has been studied. It has been proved that the strain gradient theory considering the influence of high order deformation can explain this size effect. However, in the existing strain gradient theory, some scale parameters are not independent, some have introduced improper equilibrium conditions and some are similar to theory. The size effect can not be true and reflects the actual situation reasonably. Therefore, it is an important research content to study the mechanical behavior of the micro component, whether it is from the development of the strain gradient theory, or to accurately describe the mechanical behavior of the micro component, and to study the mechanical properties of the microstructures, including the mechanical and electrical coupling properties. The main contents include: developing the symmetric / antisymmetric and hydrostatic / partial decomposition method of strain gradient tensor, decomposing the strain gradient tensor into two forms of independent component. The constitutive equation of strain gradient elastic theory is reconstructed with the strain gradient independent component, and the isotropic elastic tensor is constructed. The constraint relation between the theory and the elastic scale tensor proves that only 3 independent scale parameters have been theoretically proved, and the basic theoretical problem of the constitutive relation of strain gradient elasticity is solved. Then, the strain gradient elastic theory with independent scale parameter is proposed, and the variational principle of strain gradient theory is developed, and the theory of strain gradient theory is developed. The equation and boundary condition are controlled and the formula of strain gradient deformation in the orthogonal curvilinear coordinate system is given. Using the new developed strain gradient elasticity theory and the assumption of plane strain, the beam bending theory is developed. The axial force, shear, bending moment, high order axial force and high order bending moment are defined in the bending of the beam, and the use of these stresses is given. The equilibrium equations and boundary conditions expressed by force force are solved in two cases: one is the problem of plane strain and the other is the Bernoulli Euler beam problem. The theoretical analysis results in two cases are in agreement with the experimental results of the bending of epoxy resin and silicon cantilever beam respectively. The new theory and Aifantis single parameter theory are applied to analyze the size effects of four typical problems: rod torsion, fixed layer shear, pure bending of thin beams and spheroid expansion. By comparing the analysis results under the two theories, it shows that the new theory can describe the size effect of various deformation problems in a unified and effective way, and the single parameter is single parameter. The theory of number is limited, which reveals the necessity of multi scale parameters in the strain gradient elasticity theory. According to the laminated structure widely used in microelectromechanical systems, the displacement modes suitable for laminated microbeams and microplates are put forward respectively. The size effect of double microbeams and microplates is established by applying the newly developed displacement mode and the principle of minimum potential energy. The corresponding equilibrium equations and boundary conditions are derived, and the bending problems of simple supported double layer microbeams and four simply supported double-layer microplates are analyzed. The results show that the bending deflection and axial stress of the double-layer microbeam and the micro plate show a clear scale effect, and the thickness of a layer of micro beam or the microplate is far from the same. The deflection deformation of the double layer microbeam or microplate is close to the deflection and deformation of the single layer microbeam or the microplate when the thickness of the micro beam or microplate is larger than that of the other layer. The strain gradient elastic theory which is independent of the scale parameter is extended to the central symmetric dielectric material, and the flexure theory, which contains only 3 independent scale parameters and 2 flexural coupling parameters, is developed. The specific expression of the internal energy density function and the constitutive relation of the independent component form. The variational principle of the flexural theory of the independent component is developed, the control equation and the boundary condition are derived, and the expression of the generalized static electricity is given. The new developed flexure theory takes into account the size effect, the polarization gradient effect and the positive inverse. It is found by theoretical evidence that not all strain gradient quantities can induce polarization, but only the anti symmetric part of the expansion gradient and partial rotational gradient can induce polarization, and the symmetric part of the tensile gradient and partial rotational gradient can not induce polarization. The new developed flexo theory has been used to establish Bernoulli Europe. The balance equation and boundary condition are given for the flexural effect model of the tension beam and the Kirchhoff circular plate. The positive inverse of the cantilever beam and the simply supported axisymmetric circular plate are studied for the cantilever beam of the free end subjected to the concentrated force and the voltage acting on the upper and lower surfaces, as well as the simple supported axisymmetric circular plates subjected to the voltage action between the uniform load and the upper and lower surfaces. It is found that the positive and inverse flexure effect of the cantilever beam and the simply supported axisymmetric circular plate shows a significant dimension dependence, and decreases with the decrease of the flexure coupling coefficient, especially when the flexure coupling coefficient is zero. The strain gradient elasticity theory and the flexure electricity proposed in this paper are proposed. The theory can effectively predict the size effect of the mechanical properties of microstructures and the flexural effect of the mechanical and electrical coupling properties. The research results have important guiding significance for the design analysis, performance prediction and experimental research of MEMS products.
【學位授予單位】：山東大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：TH-39
【參考文獻】
相關(guān)期刊論文 前10條
1 李康;駱英;徐晨光;許伯強;王晶晶;桑勝;;基于BST微懸臂梁橫向撓曲電系數(shù)的測定[J];壓電與聲光;2014年05期
2 桑勝;駱英;劉軍;徐晨光;李康;王晶晶;;Ba_(0.75)Sr_(0.25)TiO_3陶瓷縱向撓曲電系數(shù)的測試[J];中國陶瓷;2014年06期
3 趙國旗;王晶晶;駱英;徐晨光;桑勝;李康;;介電材料撓曲電系數(shù)測定平臺的研制[J];實驗力學;2014年04期
4 王曉明;王飛;趙學增;景大雷;;基于Cosserat理論的四邊簡支自由振動微平板尺度效應(yīng)研究[J];固體力學學報;2012年01期
5 ;VIBRATION ANALYSIS OF MICROSCALE PLATES BASED ON MODIFIED COUPLE STRESS THEORY[J];Acta Mechanica Solida Sinica;2010年05期
6 ;Variational principles and governing equations in nano-dielectrics with the flexoelectric effect[J];Science China(Physics,Mechanics & Astronomy);2010年08期
7 趙杰;陳萬吉;冀賓;;關(guān)于兩種二階應(yīng)變梯度理論[J];力學學報;2010年01期
8 馮秀艷;郭香華;方岱寧;王自強;;微薄梁三點彎曲的尺度效應(yīng)研究[J];力學學報;2007年04期
9 康新;席占穩(wěn);;基于Cosserat理論的微梁振動特性的尺度效應(yīng)[J];機械強度;2007年01期
10 郭香華,方岱寧,李喜德;用電子散斑法對純鎳薄片彎曲變形的測量[J];力學與實踐;2005年02期
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